p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.Q8⋊9C4, C4⋊C4.12Q8, C4.24(C4×Q8), C2.12(C4×SD16), C2.4(D4.Q8), C2.3(Q8⋊Q8), (C2×C4).111SD16, C23.777(C2×D4), (C22×C4).692D4, C22.157(C4×D4), C22.62(C4○D8), C22.4Q16.8C2, (C22×C8).43C22, C4.12(C42.C2), C22.62(C2×SD16), C4.14(C42⋊2C2), C4.23(C42⋊C2), C22.81(C8⋊C22), (C2×C42).292C22, C22.76(C22⋊Q8), C2.12(SD16⋊C4), (C22×C4).1375C23, C2.4(C23.46D4), C2.5(C23.20D4), C22.71(C8.C22), C23.65C23.5C2, C22.7C42.32C2, C22.89(C22.D4), C2.10(C23.63C23), (C4×C4⋊C4).15C2, C4⋊C4.77(C2×C4), (C2×C8).110(C2×C4), (C2×C4).270(C2×Q8), (C2×C4.Q8).16C2, (C2×C4).572(C4○D4), (C2×C4⋊C4).770C22, (C2×C4).393(C22×C4), SmallGroup(128,651)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.Q8⋊9C4
G = < a,b,c,d | a4=d4=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, cbc-1=b3, dbd-1=ab3, cd=dc >
Subgroups: 220 in 115 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.65C23, C2×C4.Q8, C4.Q8⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.63C23, C4×SD16, SD16⋊C4, Q8⋊Q8, D4.Q8, C23.46D4, C23.20D4, C4.Q8⋊9C4
►Regular action on 128 points
Generators in S128
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 111 13 107)(10 112 14 108)(11 105 15 109)(12 106 16 110)(17 104 21 100)(18 97 22 101)(19 98 23 102)(20 99 24 103)(25 96 29 92)(26 89 30 93)(27 90 31 94)(28 91 32 95)(33 73 37 77)(34 74 38 78)(35 75 39 79)(36 76 40 80)(41 81 45 85)(42 82 46 86)(43 83 47 87)(44 84 48 88)(49 68 53 72)(50 69 54 65)(51 70 55 66)(52 71 56 67)(113 121 117 125)(114 122 118 126)(115 123 119 127)(116 124 120 128)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 107 63 15)(2 110 64 10)(3 105 57 13)(4 108 58 16)(5 111 59 11)(6 106 60 14)(7 109 61 9)(8 112 62 12)(17 87 102 41)(18 82 103 44)(19 85 104 47)(20 88 97 42)(21 83 98 45)(22 86 99 48)(23 81 100 43)(24 84 101 46)(25 79 94 33)(26 74 95 36)(27 77 96 39)(28 80 89 34)(29 75 90 37)(30 78 91 40)(31 73 92 35)(32 76 93 38)(49 115 66 121)(50 118 67 124)(51 113 68 127)(52 116 69 122)(53 119 70 125)(54 114 71 128)(55 117 72 123)(56 120 65 126)
(1 78 127 103)(2 37 128 23)(3 80 121 97)(4 39 122 17)(5 74 123 99)(6 33 124 19)(7 76 125 101)(8 35 126 21)(9 32 70 84)(10 90 71 43)(11 26 72 86)(12 92 65 45)(13 28 66 88)(14 94 67 47)(15 30 68 82)(16 96 69 41)(18 63 40 113)(20 57 34 115)(22 59 36 117)(24 61 38 119)(25 50 85 106)(27 52 87 108)(29 54 81 110)(31 56 83 112)(42 105 89 49)(44 107 91 51)(46 109 93 53)(48 111 95 55)(58 77 116 102)(60 79 118 104)(62 73 120 98)(64 75 114 100)
G:=sub<Sym(128)| (1,57,5,61)(2,58,6,62)(3,59,7,63)(4,60,8,64)(9,111,13,107)(10,112,14,108)(11,105,15,109)(12,106,16,110)(17,104,21,100)(18,97,22,101)(19,98,23,102)(20,99,24,103)(25,96,29,92)(26,89,30,93)(27,90,31,94)(28,91,32,95)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,81,45,85)(42,82,46,86)(43,83,47,87)(44,84,48,88)(49,68,53,72)(50,69,54,65)(51,70,55,66)(52,71,56,67)(113,121,117,125)(114,122,118,126)(115,123,119,127)(116,124,120,128), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,107,63,15)(2,110,64,10)(3,105,57,13)(4,108,58,16)(5,111,59,11)(6,106,60,14)(7,109,61,9)(8,112,62,12)(17,87,102,41)(18,82,103,44)(19,85,104,47)(20,88,97,42)(21,83,98,45)(22,86,99,48)(23,81,100,43)(24,84,101,46)(25,79,94,33)(26,74,95,36)(27,77,96,39)(28,80,89,34)(29,75,90,37)(30,78,91,40)(31,73,92,35)(32,76,93,38)(49,115,66,121)(50,118,67,124)(51,113,68,127)(52,116,69,122)(53,119,70,125)(54,114,71,128)(55,117,72,123)(56,120,65,126), (1,78,127,103)(2,37,128,23)(3,80,121,97)(4,39,122,17)(5,74,123,99)(6,33,124,19)(7,76,125,101)(8,35,126,21)(9,32,70,84)(10,90,71,43)(11,26,72,86)(12,92,65,45)(13,28,66,88)(14,94,67,47)(15,30,68,82)(16,96,69,41)(18,63,40,113)(20,57,34,115)(22,59,36,117)(24,61,38,119)(25,50,85,106)(27,52,87,108)(29,54,81,110)(31,56,83,112)(42,105,89,49)(44,107,91,51)(46,109,93,53)(48,111,95,55)(58,77,116,102)(60,79,118,104)(62,73,120,98)(64,75,114,100)>;
G:=Group( (1,57,5,61)(2,58,6,62)(3,59,7,63)(4,60,8,64)(9,111,13,107)(10,112,14,108)(11,105,15,109)(12,106,16,110)(17,104,21,100)(18,97,22,101)(19,98,23,102)(20,99,24,103)(25,96,29,92)(26,89,30,93)(27,90,31,94)(28,91,32,95)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,81,45,85)(42,82,46,86)(43,83,47,87)(44,84,48,88)(49,68,53,72)(50,69,54,65)(51,70,55,66)(52,71,56,67)(113,121,117,125)(114,122,118,126)(115,123,119,127)(116,124,120,128), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,107,63,15)(2,110,64,10)(3,105,57,13)(4,108,58,16)(5,111,59,11)(6,106,60,14)(7,109,61,9)(8,112,62,12)(17,87,102,41)(18,82,103,44)(19,85,104,47)(20,88,97,42)(21,83,98,45)(22,86,99,48)(23,81,100,43)(24,84,101,46)(25,79,94,33)(26,74,95,36)(27,77,96,39)(28,80,89,34)(29,75,90,37)(30,78,91,40)(31,73,92,35)(32,76,93,38)(49,115,66,121)(50,118,67,124)(51,113,68,127)(52,116,69,122)(53,119,70,125)(54,114,71,128)(55,117,72,123)(56,120,65,126), (1,78,127,103)(2,37,128,23)(3,80,121,97)(4,39,122,17)(5,74,123,99)(6,33,124,19)(7,76,125,101)(8,35,126,21)(9,32,70,84)(10,90,71,43)(11,26,72,86)(12,92,65,45)(13,28,66,88)(14,94,67,47)(15,30,68,82)(16,96,69,41)(18,63,40,113)(20,57,34,115)(22,59,36,117)(24,61,38,119)(25,50,85,106)(27,52,87,108)(29,54,81,110)(31,56,83,112)(42,105,89,49)(44,107,91,51)(46,109,93,53)(48,111,95,55)(58,77,116,102)(60,79,118,104)(62,73,120,98)(64,75,114,100) );
G=PermutationGroup([[(1,57,5,61),(2,58,6,62),(3,59,7,63),(4,60,8,64),(9,111,13,107),(10,112,14,108),(11,105,15,109),(12,106,16,110),(17,104,21,100),(18,97,22,101),(19,98,23,102),(20,99,24,103),(25,96,29,92),(26,89,30,93),(27,90,31,94),(28,91,32,95),(33,73,37,77),(34,74,38,78),(35,75,39,79),(36,76,40,80),(41,81,45,85),(42,82,46,86),(43,83,47,87),(44,84,48,88),(49,68,53,72),(50,69,54,65),(51,70,55,66),(52,71,56,67),(113,121,117,125),(114,122,118,126),(115,123,119,127),(116,124,120,128)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,107,63,15),(2,110,64,10),(3,105,57,13),(4,108,58,16),(5,111,59,11),(6,106,60,14),(7,109,61,9),(8,112,62,12),(17,87,102,41),(18,82,103,44),(19,85,104,47),(20,88,97,42),(21,83,98,45),(22,86,99,48),(23,81,100,43),(24,84,101,46),(25,79,94,33),(26,74,95,36),(27,77,96,39),(28,80,89,34),(29,75,90,37),(30,78,91,40),(31,73,92,35),(32,76,93,38),(49,115,66,121),(50,118,67,124),(51,113,68,127),(52,116,69,122),(53,119,70,125),(54,114,71,128),(55,117,72,123),(56,120,65,126)], [(1,78,127,103),(2,37,128,23),(3,80,121,97),(4,39,122,17),(5,74,123,99),(6,33,124,19),(7,76,125,101),(8,35,126,21),(9,32,70,84),(10,90,71,43),(11,26,72,86),(12,92,65,45),(13,28,66,88),(14,94,67,47),(15,30,68,82),(16,96,69,41),(18,63,40,113),(20,57,34,115),(22,59,36,117),(24,61,38,119),(25,50,85,106),(27,52,87,108),(29,54,81,110),(31,56,83,112),(42,105,89,49),(44,107,91,51),(46,109,93,53),(48,111,95,55),(58,77,116,102),(60,79,118,104),(62,73,120,98),(64,75,114,100)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | SD16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C4.Q8⋊9C4 | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C23.65C23 | C2×C4.Q8 | C4.Q8 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 1 | 1 |
Matrix representation of C4.Q8⋊9C4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,5,0,0,0,0,7,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,1,0,0,0,0,1,7,0,0,0,0,0,0,13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0] >;
C4.Q8⋊9C4 in GAP, Magma, Sage, TeX
C_4.Q_8\rtimes_9C_4
% in TeX
G:=Group("C4.Q8:9C4"); // GroupNames label
G:=SmallGroup(128,651);
// by ID
G=gap.SmallGroup(128,651);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,394,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=a*b^3,c*d=d*c>;
// generators/relations